Maximum Entropy Empirical Likelihood Methods Based on Bivariate Laplace Transforms and Moment Generating Functions

Maximum Entropy Empirical Likelihood (MEEL) methods are extended to bivariate distributions with closed form expressions for their bivariate Laplace transforms (BLT) or moment generating functions (BMGF) without closed form expressions for their bivariate density functions which make the implementation of the likelihood methods difficult. These distributions are often encountered in joint modeling in actuarial science and finance. Moment conditions to implement MEEL methods are given and a bivariate Laplace transform power mixture (BLTPM) is also introduced, the new operator generalizes the existing univariate one in the literature. Many new bivariate distributions including infinitely divisible(ID) distributions with closed form expressions for their BLT can be created using this operator and MEEL methods can also be applied to these bivariate distributions.


Introduction
Bivariate distributions are useful for joint modelling and naturally fitting these distributions is a necessity for pricing in insurance and finance.For example, in finance fitting a bivariate jump diffusion model to joint returns data allowed us to price a basket option accordingly, see Ruijter and Oosterlee [1] (p.B658) for a bivariate jump diffusion model.The authors advocated a two dimensions cosine method for pricing basket option.In actuarial sciences, one might consider modeling two claim amounts from an accident, i.e., the amount from body damage and the amount from material damage which are incurred simultaneously in a car accident, see Partrat [2] (p.225).Joint survival analysis is also useful for lifetime study, see Crowder [3] (pp. 121-136).
In order to find suitable bivariate parametric models, compound methods and copulas are often used.New bivariate distributions created using the traditional distribution or survival copulas are often continuous with closed form distribution functions or density functions so that methods based on likelihood functions can be applied for statistical inferences.It is also well known that many useful bivariate infinitely divisible distributions do not have closed form density function nor distribution function yet very useful for applications.
For the univariate case, the LT power mixture (LTPM) operator as introduced by Abate and Whitt [4] (pp. 92-93) for the univariate set up for creating new distribution using Laplace transforms (LT) can be used to generate many distributions with closed form LT. Furthermore, it can also be used to generate distribution which is infinitely divisible.In this paper we shall generalize the LTPM to a bivariate version, the BLTPM operator and show that the BLTPM operator can be used to generate new bivariate distributions with closed from BLT.Furthermore, the traditional survival copulas such as the Clayton copula, see Shih and Louis [5] can be used as a LT copula if the property of complete monotonicity of two specific related functions are satisfied.For another class of survival copulas, see Crowder [3] (pp. 121-138).Consequently, some distribution or survival function copulas which are defined using a generator based on a LT of a nonnegative random variable can be used to generate new distribution with prescribed marginals specified by their marginal LTs.A similar bivariate PM operator based on distribution functions or survival functions, the BDSPM operator has been introduced by Marshall and Olkin [6] (pp.834-836) to create new distributions functions and with frailty induced distributions, it also related to a class of distribution or survival Copula functions defined with a generator.The BLTPM operator can be used to generate bivariate infinite divisible distributions with closed form BLTs. It appears to be useful to have bivariate infinite divisible distributions for joint modeling as they are related to corresponding bivariate Lévy processes with stationary and independent increments.These types of processes are useful as they often lead to elegant results in risk theory in actuarial sciences and in finance.
It appears natural to extent inferences based on bivariate maximum entropy empirical likelihood (MEEL) to distributions with closed form bivariate LT (BLT) or closed form bivariate moment generating function(BMGF) which are similar to the univariate case as given by Luong [7].Beside only the BLT or BMGF is needed without asking the explicit expression of the Bivariate distribution function or density function, MEEL methods appear to be practical and the methods also offer simultaneously a goodness of fit test statistics which follows a chi-square distribution which is relatively simple to implement and appear to be useful for practitioners.Along  The asymptotic properties which already appear in the literature are restated emphasizing bases and projections of the score functions.We also discuss numerical implementation of the MEEL methods using penalty function approach as given by Luong [7].Section 5 illustrates the use of the methods by conducting a limited simulation study with a bivariate compound Poisson model as proposed by Partrat [2] (pp.220-223).With the range of parameters as considered, we observe that the MEEL estimators are two to four time more efficient than MM estimators and at the same time provide a chi-square goodness of fit test statistics for model selection.A base of nine elements is chosen in the study and there is no major numerical difficulty encountered on implementation using the penalty function approach.The methods avoid the use of the covariance matrix of the elements of the base as required for the optimum generalized moment methods (GMM) as discussed in Luong [7], the expressions for this optimum covariance matrix can be complicated.
In the next section, we shall give some examples to illustrate that in many situations we end up working with bivariate distributions with closed form BLT or closed from BMGF but without a closed form for distributions functions or density functions.

Some Examples
Example 1 The following bivariate distribution introduced by Partrat [ Poisson distribution and has closed form BLT and suitable for modelling two claim amounts from two types of claim in actuarial sciences.The vector of claim amount will have two components and each component can be expressed as a random sum as in the univariate case but the two components will be correlated due to the two numbers of claims given by the vector ( ) and follow a bivariate Poisson distribution.More precisely, the claim amount is a vector ( ) . Each of the components of X can be expressed each as a random sum.Note that dependent and identically distributed as 0 U ≥ which is a nonnegative con- tinuous random variable with LT, ( ) are independent and identically distributed as 0 V ≥ which is a nonnegative continuous random variable with LT, ( ) , N N are assumed to be independent and with other related independence assumptions as given by H1-H7 in Partrat [2] (pp.220-221) leads to the BLT of the vector X expressible as The bivariate probability generating function (BPGF) for ( ) , N N N ′ = follows a bivariate Poisson distribution then its PGF is given as ( ) ( ) ( ) ( ) The parameters 1 2 12 ф ,ф ,ф are all nonnegative.This is the only bivariate Poisson distribution which is infintely divisble(ID), see Dwass and Teicher [10].
Now if U and V are also ID then with property of the bivariate Poisson PGF, it can be seen that is a BLT for each positive integer n, using the remarks given by Abate and Whitt [4] based on results given by Feller [11]  , , This is a jump-diffusion model with the diffusion part given by B and the jump part by the random sum ∑ J .Also, the J i 's, B, Z and N are independent with N following a Poisson distribution with parameter λ.Comparing to a bivariate normal model, the bivariate jump diffusion model has an extra jump component.
The domain of ( ) , M s t X is the entire plane due to the use of the Poisson and normal distributions.Clearly, this an ID BMGF and the corresponding jump diffusion process is a bivariate Lévy process.Also, the first two moments of the vector X can be extracted from the BMGF.For this model, MEEL methods can be used for estimation and model testing.The class of bivariate normal mean variance mixture as described by McNeil et al [12] (pp.77-78) is another example where bivariate MEEL methods might be suitable.Furthermore, MEEL methods can also be used for testing the null composite hypothesis which specifies that the parametric model fits the data.The test statistics based on MEEL methods can be constructed in a unified way and obtained simultaneously with the estimation procedures.This feature is useful for doing applied works and the test statistics is less complicated than statistics based on the Mahanolobis distance, see Mc Assey [13], Muldhokar et al. [14] for statistics based on such a distance.The unique asymptotic chi-square distribution that the statistics follows across the composite hypothesis make them suitable statistics to replace the traditional chi-square test statistics as proposed by Moore and Stubblebine [15] which require closed form bivariate density functions.Model testing procedure is easy to implement if it is based on a statistics with a unique asymptotic distribution for all 0 ∈ Ω β , the parameter space is denoted by Ω.The main requirement is the model BLT has a closed form expression.These properties of bivariate MEEL methods are similar to properties of univariate MEEL methods as discussed by Luong [7] and will be further discussed in section 4. Note that asymptotic theory for empirical likelihood methods as developed by Qin and Lawless [16] (p302-308) is not restricted to univariate distribution and the same can be said for the maximum entropy version.In fact, asymptotic theory for empirical likelihoods is well established for model with multivariate observations which are independent and identically distributed using a set of fixed moment conditions.The example 2 as given by Qin and Lawless [16] (pp.311-312) is an example of the use of empirical likelihood methods for a bivariate model.In practice, we would like to choose the moment conditions so that high efficiencies can be achieved and handle the procedures numerically.We focus on these points for a class of distributions with closed form BLT or BMGF and do not emphasize asymptotic theory in the paper.

The BLTPM Operator
The Laplace transform power mixture operator (LTPM) has been introduced in the literature for creating univariate non-negative distribution and can be viewed as a continuous type of compounding operator and it makes use of LT of a dis- For illustrations, some examples of new bivariate distribution specified by BLT and created using the BLTPM operator will be given for illustrations in section 3.2.The BLTPM operator can also be used to create new distribution with one marginal distribution which is discrete and the other marginal being continuous.These types of bivariate distributions appear to be useful for actuarial science and possibly for other fields as well.
Often, we need to know whether a function can be considered as LT of some univariate random variable, the notion of completely monotonicity of a function is useful for characterizing a function to be a proper LT of a random variable and can be found in Feller [11] (pp. 439-440) but it is reproduced here for completeness as Definition and Theorem below.
n of all order and ( ) ( ) ( ) ф s with the domain given by 0 s ≥ is the LT of a random vari- able if and only if it is CM.Now if we assume that the LT of a random variable is ( ) Using the LT of α, it can be re-expressed as Observe that if A form of bivariate PM operator, the BSDPM has been used in the literature but with the use of distribution functions or survival functions, see the seminal works of Marshall and Olkin [6] and a class of survival or distribution copula functions are also obtained.This class is defined using a generator which is based on the LT of some nonnegative random variable which forms a subclass of the Archimedean class and will be further discussed in section 3.2.Due to the analogy between the BLTPM operator and BDSPM operator, these distribution or survival copula functions subject to some conditions are also LT copulas.For the Archimedean class, see Klugman et al. [17] (pp.187-213).
In section 3.2, we shall see that additional requirements are needed when working with LT's instead of distribution functions or survival functions for the use of these copulas traditionally used to create new bivariate distribution function using prescribed marginal distributions.This is due to the conditions

Example 3
For modeling two type of claims in insurance for one period of time, we might want to construct a joint continuous model using the BLTPM operator with the mixing random variable following a gamma distribution with LT ( ) ( ) ( ) ĝ t are LT of inverse Gaussian distributions with LTs given respectively by ( ) and ( ) , the Pearson product moment correlation coefficient can be obtained and the coefficient can be used to study the dependence between Y and Z.
Other depence measures can also be used to study associativity and dependence, see Chapter 4 of the book by Balakrishnan and Lai [19] (pp.141-173) and for total positivity dependence, see Barlow and Proschan [20] (pp.142-150) but we do not go into details into these dependence studies for distributions created using the BLTPM operator as the main focus of the paper are on statistical inference methods using BLTs or BMGFs.

Example 4
Bivariate mixed models with one marginal for counts and the other one for claim amounts appear to be useful in actuarial science.These models can also be constructed using the BLTPM operator.For example, let the random vector ( ) with Y being discrete and Z being continuous and nonnegative.We might specify ( ) ( ) which is the LT of a Poisson random variable with 0 λ > , let and ( ) , θ and 0 µ > which is the LT of an inverse Gaussian distribution.
The BLT of the random vector ( ) Setting 0 t = we obtain the marginal LT of Y which is given by ( ) ( ) ( ) for the PGF of a negative binomial distribution.
The marginal ( ) is the LT of a continuous random variable.

Example 5
In this example we shall obtain a bivariate negative binomial distribution using the BLTPM operator.Let ( ) ( ) with BLT given by ( ) , 1 e 1 e 1 This is a BLT of a bivariate negative binomial distribution which is also ID.In the literature, this distribution has been constructed using various methods and by many authors, see Mardia [21] (p.84).We can see that the BLTPM operator is very useful for constructing bivariate distribution with the property being ID.This property is useful in finance and actuarial sciences as the corresponding bivariate process is a bivariate Lévy process with stationary and independent increments.This property often leads to nice results in risk theory and pricing formulas in finance.Simulation of sample paths from these processes are also simplified using this property.
We briefly mention on how to simulate from bivariate distribution with a specified ( ) , L s t X created by the BLTPM operator.The procedures are similar to the procedures described in section 5 and given by Marshall and Olkin [6] (p. 840) but the inverse transform method might not be necessary explicitly.
The main steps are: 1) Generate an observation for the mixing random variable α from the distribution H which has LT ( ) Use the observed value α, generate an observation ( ) X α from a distribution with LT ( ) Often from the expression of ( ) , we have a procedure to simulate from this distribution and the inverse method might not be needed explicitly at this step.
3) Use the same observed value α, simulate another observation ( ) Y α which is independent of ( ) X α obtained from 2) from a distribution with LT ( ) The pair of observations

LT Copulas
Observe that the BLTPM operator given by expression ( 2) is similar to the BDSPM operator given by expression 2.2 in Marshall and Olkin [6] (p834) where a class of distribution or survival copulas can be obtained, it is considered below.Without loss of generality we consider distribution copulas which are related to the BDSPM operator.By using the specified marginal distributions 1 F and 2 F , a new bivariate distribution ( ) , F x y can be constructed with these two specified mar- ginals which is given by ф is a univariate LT and its inverse given by If we let ( ) which is a LT of a gamma distribution, its inverse is given by − , then we will have the classical distribution or survival distribution Clayton copula as shown in example 4.2 given by Marshall and Olkin [6] (p. 837).For other copulas of the form given by expression (4), see Shih and Louis [5].
For new bivariate distribution constructed using the BLTPM, we restrict the which is similar to the procedure as given by expression (4) using distribution or survival functions.This also means that some of the distribution or survival copulas of the class given by expression (4) can also be used as LT copulas.But the drawback of this approach using LT copula here is even by specifying the marginals ( ) It is also more difficult to simulate from a BLT ( ) , L s t X obtained using the LT copula as given by expression (5).Despite the algorithm used to generate a pair of observations from a specified BLT procedure is already given as above but when apply the procedures here we encounter the difficulty on how to simulate from distributions with LTs given by ( ) ) might not be able to recognize these distributions.The inverse transform method can be applied by using the approximate quantile function based on moment generating functions ( ) to obtain a simulated sample with some accuracy.The approximate quantile functions based on moment generating functions can be obtained explicitly using the saddle point technique and it is given by Arevelillo [23].If these univariate MGFs do not exist, these LTs can be converted to characteristic functions and the numerical methods based on the inverse method and the approximate quantile functions as described by Glasserman and Liu [24] (pp.1613-1615) can also be used to generate a pair of observations with some degree of accuracies.
There is a vast literature on survival or distribution copula, we just mention a few here.For a good general review on distribution or survival copula models with emphasis on goodness-of-fit test statistics, see Genest et al. [25] and for a good review of inference methods based on survival copulas used in actuarial science, see Klugman et al. [17] (pp.187-213), Frees and Valdez [26].
As mentioned earlier, asymptotic theory for Maximum entropy empirical likelihood (MEEL) and empirical likelihood (EL) for models where we have 1 , , n X X a sample of multivariate observations independent and identically distributed (iid) as X which follows a d-variate multivariate distribution are well established but assuming a set of moment conditions or constraints has been chosen, see Qin and Lawless [16], Schennach [27], Owen [28], Mittelhammer et al. [29]; also see discussions in section 3.2 by Luong [7] (pp.

471-472).
For applications, the question on how to choose moment conditions or equivalently bases so that MEEL methods have high efficiencies is a relevant one and we would like to address mainly this issue here for models with closed form BLTs or BMGFs as introduced in previous sections.The moment conditions are viewed as constraints and can be identifed with elements of a chosen basis used to project the true score functions as in the univariate case, see Luong [7] (pp.461-467).
Beside estimation, empirical likelihood type methods also give a chi-square test statistics for model testing which is useful for practical applications.We shall emphasize moment conditions and focus on how to choose constraints for models with closed form BLTs or BMGFs in the next section as asymptotic theory is similar to the univariate case and already discussed in section 3.2 as given by Luong [7].

MEEL Methods
Let 1 , , n X X be sample of bivariate observations and they are independent and identically distributed as , F β has no closed from but its BLT or BMGF has a closed form expression and given respectively by where these functions ( ) are unbiased estimating functions with the property , these functions also form a basis ( β used to project the true score functions and MEEL estimators can be viewed as equivalent to quasilikelihood estimators based on the projected score functions, see Luong [7] (pp.466-468).Therefore, the constraints are linked to a basis and the ability to obtain projected score functions close to the true score functions for some restricted but useful parameter space will make MEEL estimators have high efficiencies in practice.

Choice of Bases
In this section, we shall recommend a base when using a model BLT to extract moment conditions which can be used as a guideline for forming other bases if needed for applying bivariate MEEL methods.
We shall define the first five g i 's as follows which make use of the first two moments of X, i.e., ( , .
The variances of y and z are given respectively by With all these g i 's, the basis can be formed and given by LT k B g x g x k Therefore, if the number of parameters m in the model, m k < , bivariate MEEL methods can be used for estimation and model testing using this basis to generate constraints.Note that the choice of ( ) do not put restrictions on the parameter space as the model BLT is well defined on the nonnegative quadrant for all values of the vector ( ) . This might not be the case with the use of the BMGF ( ) where the domain of s might depend on β .We have to be ensured that if restrictions are imposed, the restricted parameter space is all we need for applications.The choice of points are based on the intuitve ground that the BLT contains more information at points near the origin and obviously there is some arbitrairiness on the choice of points or equivalently moment conditions for using MEEL methods.

Estimation and Model Testing
Asymptotic properties for Multivariate MEEL methods and for EL methods for models with independent and identically distributed of vector of observations are well established, see Qin and Lawless, Mittelhammer et al. [29] (p.323), Schennach [27].Now if we use the constraints as defined using a basis estimators are obtained as the vector β which maximizes the entropy or mini- mizes ( ) ( ) ( ) The λ j 's depend on β and are only implicitly defined using expression (11).
Using standard regularity conditions as indicated in the references just listed, also see section 3.2 given by Luong [7] (pp.471-472), we then have consistency and asymptotic normality for the MEEL estimators given by β , i.e., 0 ˆp  → β β , 0 β is the vector of the true parameters, ( ) , we have another consistent estimator for Ω .The asymptotic results are very similar to the ones for the univariate case as given by Luong [7] but for completeness they are reproduced to make the paper easier to follow.
For model testing, it is viewed as testing the validity of the moment conditions, i.e., tesing the null hypothesis .(13) This test might be useful for testing bivariate normality in empirical finance and it is based on the MEEL estimators β and since the basis used spans the true score functions of the bivariate normal model which makes β as efficient as the maximum likelihood estimator ML β , as the projected score functions are iden- tical to the score functions.Furthermore, the asymptotic chi-square distributions are practical for the use of these test statistics and we do not need intensive simulations to implement model testing procedures as in other methods.Also, MEEL methods can be applied for model with a singular part in the domain provided that the model BLT has a closed form expression.

Numerical Implementations
As for the univariare case, a penalty approach can be used which means that we can perform unconstrained minimization using the following objective function with respect to ( )

Conclusion
MEEL estimators using the proposed moment conditions or bases appear to have the potential of higher efficiencies than MM estimators in general.The methods also provide chi-square goodness-of-fit test statistics for model testing.These features make the methods useful for inferences for bivariate distributions with closed form BLTs or BMGFs without closed form for the bivariate density functions.For these distributions, the implementation of likelihood methods might be difficult.
Definition A function ( ) ф s with the domain given by is continuous and ID then the new bivariate distribution created is also ID.We can interpret this property using Lévy processes, as two independent Lévy processes subject to the same time change Lévy process is a bivariate Lévy process.This generalizes the argument used for the A. Luong DOI: 10.4236/ojs.2018.82017271 Open Journal of Statistics univariate case given by Abate and Whitt [4] (p. 92).

>
are proper LT are more difficult to verify than the corresponding conditions, distribution functions when the BDSPM operator is used, see Marshall and Olkin[6] (p.834).We shall give a few examples below to illustrate the use of the BLTPM operator to create bivariate ID distribution using the LT of a gamma mixing random variable.The same procedure can be used with other mixing random variables such as the inverse Gaussian random variable or the exponential mixture inverse Gaussian (EMIG) random variable.The EMIG distribution as studied by Abate and Whitt[4] (pp.95-96) is also ID, continuous and nonnegative.This distribution has not received much attention in the statistical literature despite it is used frequently in queueing theory.

.
One can recognize that this is a LT of a negative binomial distribution and for this distribtion, the corresponding probability generating function (PGF) et al.[17] (p.83)

.
The true vector of parameters is denoted by 0 ∈ Ω β .The parameter space Ω is assumed to be compact.Suppose that we can identify k constraints of the form ( ) the univariate case, the expectations are under the true parametric model.The following test statistics given below is a chi-square test statistics which has an asymptotic chi-square distribution with r k p = − degree of freedom, i.e., [7] same vein of univariate MEEL methods, bivariate MEEL methods of estimation and tests are based on constraints specified by moments be identified with elements of a basis where the true score functions are projected as in the univariate set up, see Luong[7](pp.463-465); these moment conditions which define elements of the base can be extracted from the model BLT or BMGF.
DOI: 10.4236/ojs.2018.82017266 Open Journal of Statistics conditions which can but with closed form BLTs. The BLTPM operator is introduced in Section 3 and it is shown that it can be used to create bivariate ID distribution with closed form BLT.MEEL methods are introduced in Section 3 with the proposal of two bases to generate moment conditions.The elements of the bases are based or BLT or BMGF and do not need the density functions explicitly.These bases are proposed to balance efficiencies and numerical tractability as a base with a large number of elements tend to create numerical difficulties on implementation of the MEEL methods.
also suppose that the LT of another random variable Y is ( ) α which is the LT of a random variable denoted by( )Y α are independent but when integrate out with respect to the distribution of α will give a new bivariate LT for a vector ( ) If we use moment conditions based on the BMGF such as estimation for the It appears that such a base gives a good balance between efficiency and numerically tractability for the MEEL methods and can be used as a guideline to form other bases if necessary in practice.
we do not have large computing resources but it does confirm the asymptotic results on efficiencies of MEEL estimators.More works need to be done for assessing the efficiencies of MEEL methods by using various parametric models especially in finite samples.